Ray Tracing
William Ribarsky
(modified by Amos Johnson)
1
Ray Tracing
• Trace light rays that enter eye through view-plane back from
whence they came. (# rays == # pixels)
• Use illumination model to determine color at intersection of ray
and object.
• Shadows “happen”.
Light Source
Shadow
Ray
Center of
Projection
Projection Plane
William Ribarsky
(modified by Amos Johnson)
2
BASIC RAY TRACING
1. Find intersection of ray with nearest object in scene
(Eye ray).
2. Determine if that object is in shadow relative to
light source(s). (Shadow ray(s)).
3. Determine illumination of object from light source(s).
A LARGE part of the work is in calculating
intersections. Performance can be improved by
making intersection calculations more efficient or by
avoiding unnecessary intersection calculations.
William Ribarsky
(modified by Amos Johnson)
3
Ray/Plane Intersection
Ray is defined by R(t) = R0 + Rd*t where t > 0
R0 = Origin of ray at (x0, y0, z0)
Rd = Direction of ray [xd, yd, zd] unit vector
Plane is defined by [A, B, C, D]
Ax + By + Cz + D = 0 for a point in the plane
A2 + B2 + C2 = 1
Normal Vector, N = [A, B, C]
William Ribarsky
(modified by Amos Johnson)
4
Ray/Plane (cont.)
Substitute the ray equation into the plane equation:
A(x0 + xd*t ) + B(y0 + yd*t) + C(z0 +zd*t) + D = 0
Solve for t:
t = - (Ax0 + By0 + Cz0 + D) / ( Axd + Byd + Czd)
or
t = - ( N• R0 + D) / ( N• Rd)
Note: The normal vector of the plane should usually (except for
backface removal) be for the surface facing the ray. If it isn't
then you should reverse it.
If N• Rd > 0 Then use -[A, B, C]
William Ribarsky
(modified by Amos Johnson)
5
Ray/Plane (cont.)
• If N·Rd = 0, then the ray and plane are parallel and don’t
intersect
• To Determine if the intersection point lies in the polygon
– Project the 3D plane onto a 2D surface, by remove the dimension
with the largest absolute value in the plane equation
– Use the “inside/outside” test for 2D polygons to determine if the
point lies inside or outside.
William Ribarsky
(modified by Amos Johnson)
6
Ray/Sphere Intersection
Ray is defined by R(t) = R0 + Rd*t where t > 0
R0 = Origin of ray at (x0, y0, z0)
Rd = Direction of ray [xd, yd, zd] unit vector
A sphere with center (a, b, c) and radius r may be represented by
(x - a)2 + (y - b)2 + (z - c)2 - r2 = 0
William Ribarsky
(modified by Amos Johnson)
7
Solving for t
Substitute the basic ray equation
x = x0 + xd*t
y = y0 + yd*t
z = z0+ zd*t
into the equation of the sphere
(x0 + xd*t - a)2 + (y0 + yd*t - b)2 + (z0 +zd*t - c)2 - r2 = 0
This is a quadratic equation in t!
A*t 2 + B*t + C = 0 where
A = (xd)2 + (yd)2 + (zd)2
B = 2[xd (x0 - a) + yd (y0 - b) + zd ( z0 - c)]
C = (x0 - a)2 + (y0 - b)2 + (z0 - c)2 - r2
William Ribarsky
(modified by Amos Johnson)
8
Solving for t (cont.)
1.
2.
3.
If there are no real roots, then the ray and sphere do not intersect
If there is only one real root, then the ray grazes the sphere
If there are two real roots, then the ray intersects the sphere
–
The root that gives the smallest positive t is the closet intersection point
1.
3.
2.
William Ribarsky
(modified by Amos Johnson)
9